E(X) and Var(X) with probability calculations

5 A box contains 300 discs of different colours. There are 100 pink discs, 100 blue discs and 100 orange discs. The discs of each colour are numbered from 0 to 99 . Five discs are selected at random, one at a time, with replacement. Find

1. the probability that no orange discs are selected,
2. the probability that exactly 2 discs with numbers ending in a 6 are selected,
3. the probability that exactly 2 orange discs with numbers ending in a 6 are selected,
4. the mean and variance of the number of pink discs selected.

2. Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses

1. exactly 3 of the games,
2. fewer than half of the games. Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.
3. Calculate the mean and variance for the number of these 60 games that Bhim loses.
4. Using a suitable approximation calculate the probability that Bhim loses more than 4 games.

14 The random variable \(T\) follows a binomial distribution where $$T \sim \mathrm {~B} ( 16,0.3 )$$ The mean of \(T\) is denoted by \(\mu\).
14

1. \(\quad\) Find \(\mathrm { P } ( T \leq \mu )\).
   14
2. Find the variance of \(T\).
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